ECS 253 / MAE 253, Lecture 10 May 3, Web search and decentralized search on small-world networks

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1 ECS 253 / MAE 253, Lecture 10 May 3, 2018 Web search and decentralized search on small-world networks

2 Announcements First example of the problems introduced by using software routines rather than working with the raw data/writing code. (Matlab histogram normalization.) HW3, HW3a, HW3b are posted. Due May 15th. Office hours today 3:30-4:20pm in 3057 Kemper.

3 Search for information Assume some resource of interest is stored at the vertices of a network. Need effective search algorithms for: Web pages (static network, highly profitable) Files in a file-sharing network (dynamic network) Would like to determine rapidly where in the network a particular item of interest can be found. Worst-case scenario, search all N nodes.

4 More efficient search strategies Can the nodes be ordered in any way? Then may be able to use algorithms like binary search for an ordered list. Can nodes learn information about their neighbors? Can nodes store information about their neighbors?

5 Search on power-law random graphs Adamic, R. M. Lukose, A. R. Puniyani, and B. A. Huberman, Search in power-law networks, Phys. Rev. E, ) Modeled after file-sharing networks like GNUTELLA. Nodes can store information on neighbors (and even nextnearest neighbors). The article is a very nice example of the use of generating functions for random graphs (the topic of future lectures). Breadth-first passing always to highest-degree node possible find between O(N 2/3 ) and O(N 1/2 ).

6 Decentralized search on small-world networks What is a small world? Dates back at least to 1929, Frigyes Karinthy, Everything is different The modern world was shrinking Due to technological advances in communications and travel, friendship networks could grow larger and span greater distances Michael Gurevich, 1961 MIT PhD dissertation Manfred Kochen, 1978: For the US size population, it is practically certain that any two individuals can contact one another by means of at most two intermediaries Stanley Milgram, The Small World Problem in Psychology Today, John Guare, Six Degrees of Separation a play in 1990 and film in Watts and Strogatz, Nature 1998

7 Mathematically defining a small-world [Watts and Strogatz, Nature, 393 (1998)] Start with 1D ring of N nodes, with each node connected to it s c nearest neighbors. (All nodes start with degree c; in the picture below c = 4.) Randomly re-wire each link independently with probability p. Number of edges = nc/2. Expected number of rewired edges = ncp/2.

8 Two definitions of the SW model are essentially equivalent: (a) Original model, rewire edges with probability p. Average final node degree is c. (b) Add shortcuts with probability p, but do not remove edges. Average final node degree is c + cp.

9 Ave shortest path L(p) and clustering coefficient C(p) C(p) / C(0) L(p) / L(0) p Small-worlds have small diameter and large clustering coefficient. They are remarkably easy to generate (just a tiny p required).

10 Clustering coefficient, C i C i for a vertex i is the number of pairs of neighbors of i that are connected divided by the numbers of pairs of neighbors of i. C i = (number of links between neighbors of i, excluding i) / (total number of links that could exist between neighbors). In more formal terms: C i = 2 e jk /k i (k i 1), where e jk is the total number of links between all nodes j and k that are connected to node i (NOT including i), and k i is the degree of node i. Note the factor of 2 comes from the fact we are considering undirected edges, so the total number of edges that could exist between neighbors of i is k i (k i 1)/2. If a node is disconnected (i.e., k i = 0), then assume C i = 0 for that node.

11 Six degrees (i.e., a small world) Six Degrees of Separation 1993 Dramatic Film Six Degrees: The Science of a Connected Age book by Duncan Watts

12 Fun related websites The Oracle of Bacon The path to Kevin Bacon SixDegrees.org connecting causes and celebrities Six degrees of Kevin Garnett labs/2013/10/ six degrees of kevin garnett connect any two athletes who ve ever played.html Six degrees of NBA seperation /03/04/six-degrees-of-nba-separation/ (Blog post explaining use of Dijkstra s algorithm)

13 Navigation Clearly if central coordination, can use short paths to deliver info quickly. But, can someone living in a small world actually make use of this info and do efficient decentralized routing? Instead of designing search algorithms, given a local greedy algorithm, are there any topologies that enable O(log N) delivery times?

14 Precise topologies required [J. M. Kleinberg, Navigation in a small world, Nature, 406 (2000)] Find mean delivery time t N β, unless α = 2. Only for α = 2 will decentralized routing work, and packet can go from source to destination in O(log N) steps. For d-dimensional lattice need α = d.

15 But we know greedy decentralized routing works for human networks (c.f. Milgram s experiments six-degrees of separation [ S. Milgram, The small world problem, Psych. Today, 2, 1967.] So how do we get beyond a lattice model?

16 Navigating social networks [Watts, P. S. Dodds, and M. E. J. Newman, Identity and search in social networks, Science, 296 (2002)] [Kleinberg, Small world phenomena and the dynamics of information, in Proceedings of NIPS 2001]. Premise: people navigate social networks by looking for common features between there acquaintances and the targets (occupation, city inhabited, age,...) Brings in DATA!

17 Hierarchical social distance tree Individuals are grouped into categories along many attributes. One tree for each attribute. Trees are not the network, but complementary mental constructs believed to be at work. Assume likelyhood of acquaintance falls of exponentially with social distance.

18 Building P2P architectures Chord A Scalable Peer-to- peer Lookup Service for Internet Applications I. Stoica, R. Morris, D. Karger, F. Kaashoek, H. Balakrishnan, ACM SIGCOMM, (Cited 4825 times) Gnutella Peer-to-Peer Architecture Case Study: Gnutella Network, M Ripeanu, Proceedings of International Conference on Peer-topeer Computing, 2001.

19 Summary Web search Centralized Make use of link structure (topology) Decentralized search Efficiency/Speed depends on underlying topology Gossip algorithms (D. Kempe and J. Kleinberg): spreading shared information quickly through local exchanges (e.g., sums, local averages/consensus). Applications to P2P, sensor networks... communications... satellites

20 Other important network paradigms: Bipartite networks, trees, cliques, and cores group 1 group group 2

21 Bipartite networks Hypergraphs Trees Planar graphs Cliques Other important network paradigms This content largely from Adamic s lectures

22 Some Basic Types of Graphs Paths Stars Cycles Complete Graphs Bipartite Graphs

23 Bipartite (two-mode) networks! edges occur only between two groups of nodes, not within those groups! for example, we may have individuals and events! directors and boards of directors! customers and the items they purchase! metabolites and the reactions they participate in

24 in matrix notation! B ij! = 1 if node i from the first group links to node j from the second group! = 0 otherwise! B is usually not a square matrix!! for example: we have n customers and m products j i B =

25 going from a bipartite to a one-mode graph! Two-mode network group 1! One mode projection! two nodes from the first group are connected if they link to the same node in the second group! naturally high occurrence of cliques! some loss of information! Can use weighted edges to preserve group occurrences group 2

26 Our SWN examples are from bipartite collapse The Oracle of Bacon The path to Kevin Bacon SixDegrees.org connecting causes and celebrities Six degrees of Kevin Garnett labs/2013/10/ six degrees of kevin garnett connect any two athletes who ve ever played.html Six degrees of NBA seperation /03/04/six-degrees-of-nba-separation/ (Blog post explaining use of Dijkstra s algorithm)

27 Collapsing to a one-mode network! i and k are linked if they both link to j! P ij =! k B ki B kj i k! P = B B T! the transpose of a matrix swaps B xy and B yx! if B is an nxm matrix, B T is an mxn matrix j=1 j= B = B T =

28 Matrix multiplication! general formula for matrix multiplication Z ij =! k X ik Y kj! let Z = P, X = B, Y = B T P = = = 1*1+1*1 + 1*0 + 1*0 =

29 Collapsing a two-mode network to a one mode-network! Assume the nodes in group 1 are people and the nodes in group 2 are movies! P is symmetric! The diagonal entries of P give the number of movies each person has seen! The off-diagonal elements of P give the number of movies that both people have seen P =

30 HyperGraphs! Edges join more than two nodes at a time (hyperedge)! Affliation networks! Examples! Families! Subnetworks A B A B C D C D Can be transformed to a bipartite network 9

31 Hypergraphs beyond dyadic iteractions In ways, good models of social networks. Complex Networks as Hypergraphs Ernesto Estrada, Juan A. Rodriguez-Velazquez arxiv:physics/ , Random hypergraphs and their applications, G Ghoshal, V Zlatić, G Caldarelli, MEJ Newman, Physical Review E 79 (6), Ramanathan, R., et al. Beyond graphs: Capturing groups in networks. NetSciCom, 2011 IEEE Conference, Information Flows: A Critique of Transfer Entropies Ryan G. James, Nix Barnett, James P. Crutchfield Accepted to Physical Review Letters, April 12, See also hypergraph mining, hypergraph learning algorithms, overlapping communities, link prediction...

32 Trees! Trees are undirected graphs that contain no cycles! For n nodes, number of edges m = n-1! Any node can be dedicated as the root

33 examples of trees! In nature! trees! river networks! arteries (or veins, but not both)! Man made! sewer system! Computer science! binary search trees! decision trees (AI)! Network analysis! minimum spanning trees! from one node how to reach all other nodes most quickly! may not be unique, because shortest paths are not always unique! depends on weight of edges

34 Breadth first search : explore all the neighbors first Searching on a tree Depth first search : take a step out in hop-count each iteration

35 Planar graphs! A graph is planar if it can be drawn on a plane without any edges crossing

36 Apollonian network An undirected graph formed by a process of recursively subdividing a randomly selected triangle into three smaller triangles. A planar graph with power law degree distribution, and small world property.

37 Cliques and complete graphs! K n is the complete graph (clique) with K vertices! each vertex is connected to every other vertex! there are n*(n-1)/2 undirected edges K 3 K 5 K 8

38 The k-core and k-shell

40 k-core decomposition For visualization k-core decomposition of the Internet router level AS level e.g. Carmi et. al. PNAS A model of Internet topology using k-shell decomposition A nucleus, a fractal layer, and tendrils. in random graphs and statistical physics: K-core organization of complex networks SN Dorogovtsev, AV Goltsev, JFF Mendes Physical review letters, 2006.

ECS 253 / MAE 253, Lecture 8 April 21, Web search and decentralized search on small-world networks

ECS 253 / MAE 253, Lecture 8 April 21, 2016 Web search and decentralized search on small-world networks Search for information Assume some resource of interest is stored at the vertices of a network: Web